Starmaze Puzzle

Wizard Card  -  Volume 19  -  Mr. Wizard Number 1  -  Fri, Mar 29, 1991 7:19 PM

I've waited more than two years to present this simple stack. For reasons which even I find it difficult to explain, the starmaze puzzle has become the greatest ponarv of my career. And to celebrate its unveiling, I am offering ACTUAL MONEY as a prize to any Archipelago member who can solve the puzzle! Read on...

WHAT IT DOES

The starmaze puzzle is quite simple and not that hard to solve. I have managed to con two friends into tackling it with just a pencil and a yellow pad; in both cases it took them a little over an hour.

Each starmaze pattern consists of NINE oddly shaped "cells" arranged in a 3 by 3 grid. Each cell can be either on (black) or off (white). There are 512 different ways of combining the on and off cells, so the Starmaze puzzle stack consists of 512 cards, one for each possible pattern. The puzzle begins with pattern number 16, in which only the center cell is on.

By clicking on one of the nine cells you can cause the pattern to change. The first rule is that you can only click on black cells. So in the starting position you have only one option: clicking the center cell.

The patterns do not change at random, but instead follow a simple set of rules which depend on which cell is clicked. Here is the first of several monetary challenges:

I will send a crisp ONE DOLLAR BILL to any member who contributes a volume 20 voice card which describes, in plain English, the rules that govern pattern changes!

This is EXTREMELY easy to figure out. All you have to do is spend a few minutes clicking on the cells at random and watch how the pattern changes each time.

As you move from pattern to pattern you will notice that each pattern has a number which appears at the bottom of the screen. This raises another relatively easy challenge:

I will pay ANOTHER DOLLAR to any member who's volume 20 voice card describes HOW these patterns are numbered. Your answer should include a reference to a very famous and ancient magic square.

But these are only side issues. Once you understand how to change the patterns you are ready to SOLVE the Starmaze puzzle. The object is to find the shortest possible path from the starting position, pattern 16, to the finishing position. The finishing position is the exact inverse of the starting position, that is, every cell EXCEPT the center is black. When you reach this pattern you will hear the same haunting refrain that greets you upon entering ths stack.

I will pay FIVE DOLLARS to any member who provides a solution to the Starmaze Puzzle. Your solution should appear in the form of a list of TWELVE numbers, one for each pattern occuring on the journey from start to finish. The first number is 16. All you have to do is find the other 11 numbers!

As it turns out, there are many different paths that lead from start to finish in the minimum 11 steps. How many? You tell me!

I will pay ANOTHER FIVE DOLLARS to any member who can tell me exactly how many different 11 step paths can be found from start to finish. Your answer should include the reasoning you employed to deduce this number.

Would you like to earn EVEN MORE money? Here are several more challenges.

As you move through the starmaze, you may find yourself going in circles. You can move back to the previous pattern by clicking the right dragon button, but this is cheating. From all but FOUR of the 512 patterns it is possible to find a path that leads from the pattern back to itself in exactly FOUR moves. There is no way to achieve this feat in less than FOUR moves.

I will pay the incredible sum of TEN DOLLARS to any member who can provide the algorithm which describes how to find a path which returns to a given pattern in the minimum of FOUR steps. Your answer must also include a list of the FOUR patterns for which no such path exists. You should also speculate briefly about the significance of the number FOUR. Why does it take four steps instead of three or five or seven? What's so important about the number FOUR?

One last challenge. It turns out that this Starmaze puzzle is just a specific instance of an infinite set of n-dimensional Starmaze puzzles. In general, a proper n-dimensional starmaze puzzle consists of an even number of "Yin" cells, an even number of "Yang" cells, and a single "Center" cell. The Yin and Yang cells are arranged in a circle in a boy-girl-boy-girl arrangement. Each Yin cell affects its two Yang neighbors. Each Yang cell affects its two Yin neighbors. In addition, the Yin cells affect the Center, and the Center affects the Yang cells. The total number of cells in a puzzle indicates the "dimension" of the puzzle. Thus there is a 5 dimensional puzzle, a 9 dimensional puzzle (which is what this one is), a 13 dimensional puzzle, etc. In every case the problem is to find a path from the Center-ON position to the All-but-Center-ON position.

I will pay ANOTHER TEN DOLLARS to any member who can provide a general purpose algorithm for solving an n-dimensional Starmaze puzzle. As an illustration, you must also include a sample solution for the 25 dimensional puzzle. HINT: there are SEVEN basic steps to solving any Starmaze puzzle.

Let's review the extravagant prizes I have just offered. One dollar for the transformation rules, one dollar for the numbering scheme, five dollars for the solution, five dollars for the number of different solutions, ten dollars for the circling algorithm, and ten dollars for the general solution. To qualify, all answers must appear in a voice card contributed to VOLUME 20. The contest expires with the release of Volume 20. As an added incentive I hereby make the following UNBELIEVABLE offer:

I will pay ONE HUNDRED DOLLARS to any member who solves ALL SIX of these challenges in Volume 20!

WOW! What a sensational offer! Please push the Push Me button right now and spend a few minutes with this delightful puzzle. Incidentally, as you wander through the maze, beware of a certain pattern called the PIT. If you ever encounter it you will immediately understand how it got it's name. I won't even bother to warn you about the WOMB because you'll never find it anyway.

HOW IT WORKS

This stack is EXTREMELY simple. Essentially, it consists of 512 cards with background buttons and a single script which jumps from card to card. In fact in terms of scripting, this is the simplest Mr. Wizard stack I have ever written.

However, there was one challenge that I could not overcome until the release of HyperCard 2.0. To achieve the elegant effect of the current stack, I needed a way to implement large, irregularly shaped buttons. Unlike Plus and SuperCard, all buttons in HyperCard (even in version 2) have to be rectangular. There are some XCMDs available to get around this, but the ones I tried had undesirable side effects.

Actually, I don't really need to shape the buttons to exactly match the cells. But I DO need a clean presentation of each possible pattern. One way of doing this would be to paint each pattern separately. This is quite possible, but takes up an ENORMOUS amount of space: almost SEVEN MEGABYTES (I know because I tried it!).

So I need some way of drawing the patterns without using separate graphics for each card. The solution came with a new HyperCard 2.0 feature: the Background Button.

Background buttons allow you to create a single button that can appear hilited on one card and unhilited on the next. Through the use of customized icons, I was able to form each of the nine cells as a series of background buttons. The result is a 512 card stack that weighs in at under 250K even with sound effects, PICT resources, and other embellishments.

The nine cells of each pattern are built out of 58 special icon buttons! I won't describe what I went through in devising and placing these buttons; suffice it to say that some cursing was involved. Once the buttons were in place, all I had to do was write a one-time procedure that turned some of them on and some of them off for each card.

If you look a little closer at this stack you will notice something very strange. There are TWO different backgrounds each containing 256 cards. The only difference between the two backgrounds is that one, the "sun patterns," display WHITE dragons, while the other, the "moon patterns," display GRAY dragons. The two types of cards appear to be shuffled at random, but you may notice that regardless of how you move in the maze, you always alternate from moon to sun and back again.

One other feature that may interest junior stackheads is the scrolling PICTure that appears whenever you push the comments button in the lower right hand corner. This is another new feature of HyperCard 2.0. It is possible to display portions of arbitrarily sized pictures in a separate window, and the pictures can even be in color!

It may have occured to some of you that you could solve some of the cash challenges listed above by examining the underlying script. PLEASE DO! You will find it on the stack level, a 12 line handler called "Punch." The script is shorter than this paragraph and is the only significant script in the entire stack. Look at it all you like! I doubt if it will do you any good. The design of this stack is just as mysterious as the puzzle itself.

GOOD LUCK! Even if you don't win any prize money, I hope you will at least spend a few minutes in the maze. Get a feel for the place. Next month I will release a sequel stack that includes a MAP of the maze! Happy trails!